In a large skillet, heat the vegetable oil. Add the spice paste and cook over moderately high heat, stirring until it begins to sizzle and brown, about 3 minutes. Reduce the heat to moderate and whisk in the coconut milk, lime juice, fish sauce and brown sugar. Bring to a boil. Stir in the chopped peanuts and season the satay sauce with salt and pepper. Remove from the heat.